On nonnegative solutions for the Functionalized Cahn–Hilliard equation with degenerate mobility

• Main Authors: Shibin Dai, Qiang Liu, Toai Luong, Keith Promislow
• Format: Article
• Language: English
• Published: Elsevier 2021-11-01
• Series: Results in Applied Mathematics
• Subjects: Weak solutions; Nonnegative solutions; The Functionalized Cahn–Hilliard equation; Degenerate mobility
• Online Access: http://www.sciencedirect.com/science/article/pii/S2590037421000406

The Functionalized Cahn–Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn–Hilliard equation subject to a degenerate mobility M(u)that is zero for u≤0. Assuming the initial data u0(x)is positive, we construct a weak solution as the limit of solutions corresponding to non-degenerate mobilities and verify that it satisfies an energy dissipation inequality. Our approach is a combination of Galerkin approximation, energy estimates, and weak convergence methods.